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1 /* Copyright 2008, Google Inc. |
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2 * All rights reserved. |
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3 * |
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4 * Redistribution and use in source and binary forms, with or without |
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5 * modification, are permitted provided that the following conditions are |
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6 * met: |
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7 * |
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8 * * Redistributions of source code must retain the above copyright |
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9 * notice, this list of conditions and the following disclaimer. |
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10 * * Redistributions in binary form must reproduce the above |
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11 * copyright notice, this list of conditions and the following disclaimer |
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12 * in the documentation and/or other materials provided with the |
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13 * distribution. |
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14 * * Neither the name of Google Inc. nor the names of its |
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15 * contributors may be used to endorse or promote products derived from |
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16 * this software without specific prior written permission. |
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17 * |
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18 * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS |
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19 * "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT |
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20 * LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR |
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21 * A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT |
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22 * OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, |
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23 * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT |
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24 * LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, |
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25 * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY |
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26 * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT |
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27 * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE |
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28 * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. |
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29 * |
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30 * curve25519-donna: Curve25519 elliptic curve, public key function |
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31 * |
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32 * http://code.google.com/p/curve25519-donna/ |
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33 * |
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34 * Adam Langley <[email protected]> |
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35 * |
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36 * Derived from public domain C code by Daniel J. Bernstein <[email protected]> |
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37 * |
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38 * More information about curve25519 can be found here |
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39 * http://cr.yp.to/ecdh.html |
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40 * |
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41 * djb's sample implementation of curve25519 is written in a special assembly |
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42 * language called qhasm and uses the floating point registers. |
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43 * |
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44 * This is, almost, a clean room reimplementation from the curve25519 paper. It |
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45 * uses many of the tricks described therein. Only the crecip function is taken |
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46 * from the sample implementation. |
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47 */ |
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48 |
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49 #include <string.h> |
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50 #include <stdint.h> |
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51 |
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52 #ifdef _MSC_VER |
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53 #define inline __inline |
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54 #endif |
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55 |
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56 typedef uint8_t u8; |
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57 typedef int32_t s32; |
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58 typedef int64_t limb; |
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59 |
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60 /* Field element representation: |
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61 * |
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62 * Field elements are written as an array of signed, 64-bit limbs, least |
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63 * significant first. The value of the field element is: |
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64 * x[0] + 2^26·x[1] + x^51·x[2] + 2^102·x[3] + ... |
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65 * |
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66 * i.e. the limbs are 26, 25, 26, 25, ... bits wide. |
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67 */ |
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68 |
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69 /* Sum two numbers: output += in */ |
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70 static void fsum(limb *output, const limb *in) { |
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71 unsigned i; |
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72 for (i = 0; i < 10; i += 2) { |
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73 output[0+i] = (output[0+i] + in[0+i]); |
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74 output[1+i] = (output[1+i] + in[1+i]); |
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75 } |
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76 } |
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77 |
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78 /* Find the difference of two numbers: output = in - output |
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79 * (note the order of the arguments!) |
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80 */ |
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81 static void fdifference(limb *output, const limb *in) { |
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82 unsigned i; |
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83 for (i = 0; i < 10; ++i) { |
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84 output[i] = (in[i] - output[i]); |
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85 } |
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86 } |
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87 |
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88 /* Multiply a number by a scalar: output = in * scalar */ |
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89 static void fscalar_product(limb *output, const limb *in, const limb scalar) { |
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90 unsigned i; |
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91 for (i = 0; i < 10; ++i) { |
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92 output[i] = in[i] * scalar; |
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93 } |
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94 } |
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95 |
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96 /* Multiply two numbers: output = in2 * in |
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97 * |
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98 * output must be distinct to both inputs. The inputs are reduced coefficient |
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99 * form, the output is not. |
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100 */ |
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101 static void fproduct(limb *output, const limb *in2, const limb *in) { |
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102 output[0] = ((limb) ((s32) in2[0])) * ((s32) in[0]); |
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103 output[1] = ((limb) ((s32) in2[0])) * ((s32) in[1]) + |
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104 ((limb) ((s32) in2[1])) * ((s32) in[0]); |
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105 output[2] = 2 * ((limb) ((s32) in2[1])) * ((s32) in[1]) + |
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106 ((limb) ((s32) in2[0])) * ((s32) in[2]) + |
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107 ((limb) ((s32) in2[2])) * ((s32) in[0]); |
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108 output[3] = ((limb) ((s32) in2[1])) * ((s32) in[2]) + |
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109 ((limb) ((s32) in2[2])) * ((s32) in[1]) + |
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110 ((limb) ((s32) in2[0])) * ((s32) in[3]) + |
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111 ((limb) ((s32) in2[3])) * ((s32) in[0]); |
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112 output[4] = ((limb) ((s32) in2[2])) * ((s32) in[2]) + |
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113 2 * (((limb) ((s32) in2[1])) * ((s32) in[3]) + |
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114 ((limb) ((s32) in2[3])) * ((s32) in[1])) + |
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115 ((limb) ((s32) in2[0])) * ((s32) in[4]) + |
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116 ((limb) ((s32) in2[4])) * ((s32) in[0]); |
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117 output[5] = ((limb) ((s32) in2[2])) * ((s32) in[3]) + |
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118 ((limb) ((s32) in2[3])) * ((s32) in[2]) + |
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119 ((limb) ((s32) in2[1])) * ((s32) in[4]) + |
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120 ((limb) ((s32) in2[4])) * ((s32) in[1]) + |
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121 ((limb) ((s32) in2[0])) * ((s32) in[5]) + |
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122 ((limb) ((s32) in2[5])) * ((s32) in[0]); |
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123 output[6] = 2 * (((limb) ((s32) in2[3])) * ((s32) in[3]) + |
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124 ((limb) ((s32) in2[1])) * ((s32) in[5]) + |
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125 ((limb) ((s32) in2[5])) * ((s32) in[1])) + |
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126 ((limb) ((s32) in2[2])) * ((s32) in[4]) + |
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127 ((limb) ((s32) in2[4])) * ((s32) in[2]) + |
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128 ((limb) ((s32) in2[0])) * ((s32) in[6]) + |
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129 ((limb) ((s32) in2[6])) * ((s32) in[0]); |
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130 output[7] = ((limb) ((s32) in2[3])) * ((s32) in[4]) + |
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131 ((limb) ((s32) in2[4])) * ((s32) in[3]) + |
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132 ((limb) ((s32) in2[2])) * ((s32) in[5]) + |
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133 ((limb) ((s32) in2[5])) * ((s32) in[2]) + |
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134 ((limb) ((s32) in2[1])) * ((s32) in[6]) + |
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135 ((limb) ((s32) in2[6])) * ((s32) in[1]) + |
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136 ((limb) ((s32) in2[0])) * ((s32) in[7]) + |
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137 ((limb) ((s32) in2[7])) * ((s32) in[0]); |
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138 output[8] = ((limb) ((s32) in2[4])) * ((s32) in[4]) + |
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139 2 * (((limb) ((s32) in2[3])) * ((s32) in[5]) + |
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140 ((limb) ((s32) in2[5])) * ((s32) in[3]) + |
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141 ((limb) ((s32) in2[1])) * ((s32) in[7]) + |
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142 ((limb) ((s32) in2[7])) * ((s32) in[1])) + |
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143 ((limb) ((s32) in2[2])) * ((s32) in[6]) + |
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144 ((limb) ((s32) in2[6])) * ((s32) in[2]) + |
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145 ((limb) ((s32) in2[0])) * ((s32) in[8]) + |
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146 ((limb) ((s32) in2[8])) * ((s32) in[0]); |
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147 output[9] = ((limb) ((s32) in2[4])) * ((s32) in[5]) + |
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148 ((limb) ((s32) in2[5])) * ((s32) in[4]) + |
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149 ((limb) ((s32) in2[3])) * ((s32) in[6]) + |
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150 ((limb) ((s32) in2[6])) * ((s32) in[3]) + |
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151 ((limb) ((s32) in2[2])) * ((s32) in[7]) + |
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152 ((limb) ((s32) in2[7])) * ((s32) in[2]) + |
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153 ((limb) ((s32) in2[1])) * ((s32) in[8]) + |
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154 ((limb) ((s32) in2[8])) * ((s32) in[1]) + |
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155 ((limb) ((s32) in2[0])) * ((s32) in[9]) + |
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156 ((limb) ((s32) in2[9])) * ((s32) in[0]); |
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157 output[10] = 2 * (((limb) ((s32) in2[5])) * ((s32) in[5]) + |
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158 ((limb) ((s32) in2[3])) * ((s32) in[7]) + |
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159 ((limb) ((s32) in2[7])) * ((s32) in[3]) + |
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160 ((limb) ((s32) in2[1])) * ((s32) in[9]) + |
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161 ((limb) ((s32) in2[9])) * ((s32) in[1])) + |
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162 ((limb) ((s32) in2[4])) * ((s32) in[6]) + |
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163 ((limb) ((s32) in2[6])) * ((s32) in[4]) + |
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164 ((limb) ((s32) in2[2])) * ((s32) in[8]) + |
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165 ((limb) ((s32) in2[8])) * ((s32) in[2]); |
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166 output[11] = ((limb) ((s32) in2[5])) * ((s32) in[6]) + |
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167 ((limb) ((s32) in2[6])) * ((s32) in[5]) + |
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168 ((limb) ((s32) in2[4])) * ((s32) in[7]) + |
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169 ((limb) ((s32) in2[7])) * ((s32) in[4]) + |
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170 ((limb) ((s32) in2[3])) * ((s32) in[8]) + |
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171 ((limb) ((s32) in2[8])) * ((s32) in[3]) + |
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172 ((limb) ((s32) in2[2])) * ((s32) in[9]) + |
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173 ((limb) ((s32) in2[9])) * ((s32) in[2]); |
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174 output[12] = ((limb) ((s32) in2[6])) * ((s32) in[6]) + |
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175 2 * (((limb) ((s32) in2[5])) * ((s32) in[7]) + |
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176 ((limb) ((s32) in2[7])) * ((s32) in[5]) + |
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177 ((limb) ((s32) in2[3])) * ((s32) in[9]) + |
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178 ((limb) ((s32) in2[9])) * ((s32) in[3])) + |
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179 ((limb) ((s32) in2[4])) * ((s32) in[8]) + |
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180 ((limb) ((s32) in2[8])) * ((s32) in[4]); |
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181 output[13] = ((limb) ((s32) in2[6])) * ((s32) in[7]) + |
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182 ((limb) ((s32) in2[7])) * ((s32) in[6]) + |
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183 ((limb) ((s32) in2[5])) * ((s32) in[8]) + |
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184 ((limb) ((s32) in2[8])) * ((s32) in[5]) + |
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185 ((limb) ((s32) in2[4])) * ((s32) in[9]) + |
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186 ((limb) ((s32) in2[9])) * ((s32) in[4]); |
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187 output[14] = 2 * (((limb) ((s32) in2[7])) * ((s32) in[7]) + |
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188 ((limb) ((s32) in2[5])) * ((s32) in[9]) + |
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189 ((limb) ((s32) in2[9])) * ((s32) in[5])) + |
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190 ((limb) ((s32) in2[6])) * ((s32) in[8]) + |
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191 ((limb) ((s32) in2[8])) * ((s32) in[6]); |
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192 output[15] = ((limb) ((s32) in2[7])) * ((s32) in[8]) + |
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193 ((limb) ((s32) in2[8])) * ((s32) in[7]) + |
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194 ((limb) ((s32) in2[6])) * ((s32) in[9]) + |
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195 ((limb) ((s32) in2[9])) * ((s32) in[6]); |
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196 output[16] = ((limb) ((s32) in2[8])) * ((s32) in[8]) + |
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197 2 * (((limb) ((s32) in2[7])) * ((s32) in[9]) + |
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198 ((limb) ((s32) in2[9])) * ((s32) in[7])); |
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199 output[17] = ((limb) ((s32) in2[8])) * ((s32) in[9]) + |
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200 ((limb) ((s32) in2[9])) * ((s32) in[8]); |
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201 output[18] = 2 * ((limb) ((s32) in2[9])) * ((s32) in[9]); |
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202 } |
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203 |
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204 /* Reduce a long form to a short form by taking the input mod 2^255 - 19. */ |
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205 static void freduce_degree(limb *output) { |
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206 /* Each of these shifts and adds ends up multiplying the value by 19. */ |
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207 output[8] += output[18] << 4; |
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208 output[8] += output[18] << 1; |
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209 output[8] += output[18]; |
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210 output[7] += output[17] << 4; |
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211 output[7] += output[17] << 1; |
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212 output[7] += output[17]; |
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213 output[6] += output[16] << 4; |
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214 output[6] += output[16] << 1; |
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215 output[6] += output[16]; |
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216 output[5] += output[15] << 4; |
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217 output[5] += output[15] << 1; |
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218 output[5] += output[15]; |
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219 output[4] += output[14] << 4; |
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220 output[4] += output[14] << 1; |
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221 output[4] += output[14]; |
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222 output[3] += output[13] << 4; |
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223 output[3] += output[13] << 1; |
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224 output[3] += output[13]; |
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225 output[2] += output[12] << 4; |
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226 output[2] += output[12] << 1; |
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227 output[2] += output[12]; |
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228 output[1] += output[11] << 4; |
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229 output[1] += output[11] << 1; |
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230 output[1] += output[11]; |
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231 output[0] += output[10] << 4; |
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232 output[0] += output[10] << 1; |
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233 output[0] += output[10]; |
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234 } |
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235 |
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236 #if (-1 & 3) != 3 |
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237 #error "This code only works on a two's complement system" |
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238 #endif |
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239 |
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240 /* return v / 2^26, using only shifts and adds. */ |
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241 static inline limb |
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242 div_by_2_26(const limb v) |
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243 { |
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244 /* High word of v; no shift needed*/ |
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245 const uint32_t highword = (uint32_t) (((uint64_t) v) >> 32); |
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246 /* Set to all 1s if v was negative; else set to 0s. */ |
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247 const int32_t sign = ((int32_t) highword) >> 31; |
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248 /* Set to 0x3ffffff if v was negative; else set to 0. */ |
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249 const int32_t roundoff = ((uint32_t) sign) >> 6; |
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250 /* Should return v / (1<<26) */ |
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251 return (v + roundoff) >> 26; |
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252 } |
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253 |
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254 /* return v / (2^25), using only shifts and adds. */ |
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255 static inline limb |
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256 div_by_2_25(const limb v) |
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257 { |
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258 /* High word of v; no shift needed*/ |
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259 const uint32_t highword = (uint32_t) (((uint64_t) v) >> 32); |
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260 /* Set to all 1s if v was negative; else set to 0s. */ |
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261 const int32_t sign = ((int32_t) highword) >> 31; |
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262 /* Set to 0x1ffffff if v was negative; else set to 0. */ |
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263 const int32_t roundoff = ((uint32_t) sign) >> 7; |
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264 /* Should return v / (1<<25) */ |
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265 return (v + roundoff) >> 25; |
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266 } |
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267 |
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268 static inline s32 |
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269 div_s32_by_2_25(const s32 v) |
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270 { |
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271 const s32 roundoff = ((uint32_t)(v >> 31)) >> 7; |
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272 return (v + roundoff) >> 25; |
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273 } |
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274 |
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275 /* Reduce all coefficients of the short form input so that |x| < 2^26. |
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276 * |
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277 * On entry: |output[i]| < 2^62 |
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278 */ |
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279 static void freduce_coefficients(limb *output) { |
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280 unsigned i; |
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281 |
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282 output[10] = 0; |
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283 |
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284 for (i = 0; i < 10; i += 2) { |
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285 limb over = div_by_2_26(output[i]); |
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286 output[i] -= over << 26; |
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287 output[i+1] += over; |
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288 |
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289 over = div_by_2_25(output[i+1]); |
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290 output[i+1] -= over << 25; |
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291 output[i+2] += over; |
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292 } |
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293 /* Now |output[10]| < 2 ^ 38 and all other coefficients are reduced. */ |
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294 output[0] += output[10] << 4; |
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295 output[0] += output[10] << 1; |
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296 output[0] += output[10]; |
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297 |
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298 output[10] = 0; |
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299 |
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300 /* Now output[1..9] are reduced, and |output[0]| < 2^26 + 19 * 2^38 |
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301 * So |over| will be no more than 77825 */ |
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302 { |
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303 limb over = div_by_2_26(output[0]); |
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304 output[0] -= over << 26; |
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305 output[1] += over; |
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306 } |
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307 |
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308 /* Now output[0,2..9] are reduced, and |output[1]| < 2^25 + 77825 |
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309 * So |over| will be no more than 1. */ |
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310 { |
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311 /* output[1] fits in 32 bits, so we can use div_s32_by_2_25 here. */ |
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312 s32 over32 = div_s32_by_2_25((s32) output[1]); |
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313 output[1] -= over32 << 25; |
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314 output[2] += over32; |
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315 } |
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316 |
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317 /* Finally, output[0,1,3..9] are reduced, and output[2] is "nearly reduced": |
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318 * we have |output[2]| <= 2^26. This is good enough for all of our math, |
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319 * but it will require an extra freduce_coefficients before fcontract. */ |
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320 } |
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321 |
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322 /* A helpful wrapper around fproduct: output = in * in2. |
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323 * |
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324 * output must be distinct to both inputs. The output is reduced degree and |
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325 * reduced coefficient. |
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326 */ |
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327 static void |
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328 fmul(limb *output, const limb *in, const limb *in2) { |
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329 limb t[19]; |
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330 fproduct(t, in, in2); |
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331 freduce_degree(t); |
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332 freduce_coefficients(t); |
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333 memcpy(output, t, sizeof(limb) * 10); |
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334 } |
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335 |
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336 static void fsquare_inner(limb *output, const limb *in) { |
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337 output[0] = ((limb) ((s32) in[0])) * ((s32) in[0]); |
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338 output[1] = 2 * ((limb) ((s32) in[0])) * ((s32) in[1]); |
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339 output[2] = 2 * (((limb) ((s32) in[1])) * ((s32) in[1]) + |
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340 ((limb) ((s32) in[0])) * ((s32) in[2])); |
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341 output[3] = 2 * (((limb) ((s32) in[1])) * ((s32) in[2]) + |
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342 ((limb) ((s32) in[0])) * ((s32) in[3])); |
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343 output[4] = ((limb) ((s32) in[2])) * ((s32) in[2]) + |
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344 4 * ((limb) ((s32) in[1])) * ((s32) in[3]) + |
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345 2 * ((limb) ((s32) in[0])) * ((s32) in[4]); |
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346 output[5] = 2 * (((limb) ((s32) in[2])) * ((s32) in[3]) + |
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347 ((limb) ((s32) in[1])) * ((s32) in[4]) + |
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348 ((limb) ((s32) in[0])) * ((s32) in[5])); |
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349 output[6] = 2 * (((limb) ((s32) in[3])) * ((s32) in[3]) + |
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350 ((limb) ((s32) in[2])) * ((s32) in[4]) + |
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351 ((limb) ((s32) in[0])) * ((s32) in[6]) + |
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352 2 * ((limb) ((s32) in[1])) * ((s32) in[5])); |
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353 output[7] = 2 * (((limb) ((s32) in[3])) * ((s32) in[4]) + |
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354 ((limb) ((s32) in[2])) * ((s32) in[5]) + |
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355 ((limb) ((s32) in[1])) * ((s32) in[6]) + |
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356 ((limb) ((s32) in[0])) * ((s32) in[7])); |
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357 output[8] = ((limb) ((s32) in[4])) * ((s32) in[4]) + |
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358 2 * (((limb) ((s32) in[2])) * ((s32) in[6]) + |
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359 ((limb) ((s32) in[0])) * ((s32) in[8]) + |
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360 2 * (((limb) ((s32) in[1])) * ((s32) in[7]) + |
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361 ((limb) ((s32) in[3])) * ((s32) in[5]))); |
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362 output[9] = 2 * (((limb) ((s32) in[4])) * ((s32) in[5]) + |
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363 ((limb) ((s32) in[3])) * ((s32) in[6]) + |
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364 ((limb) ((s32) in[2])) * ((s32) in[7]) + |
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365 ((limb) ((s32) in[1])) * ((s32) in[8]) + |
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366 ((limb) ((s32) in[0])) * ((s32) in[9])); |
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367 output[10] = 2 * (((limb) ((s32) in[5])) * ((s32) in[5]) + |
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368 ((limb) ((s32) in[4])) * ((s32) in[6]) + |
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369 ((limb) ((s32) in[2])) * ((s32) in[8]) + |
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370 2 * (((limb) ((s32) in[3])) * ((s32) in[7]) + |
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371 ((limb) ((s32) in[1])) * ((s32) in[9]))); |
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372 output[11] = 2 * (((limb) ((s32) in[5])) * ((s32) in[6]) + |
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373 ((limb) ((s32) in[4])) * ((s32) in[7]) + |
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374 ((limb) ((s32) in[3])) * ((s32) in[8]) + |
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375 ((limb) ((s32) in[2])) * ((s32) in[9])); |
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376 output[12] = ((limb) ((s32) in[6])) * ((s32) in[6]) + |
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377 2 * (((limb) ((s32) in[4])) * ((s32) in[8]) + |
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378 2 * (((limb) ((s32) in[5])) * ((s32) in[7]) + |
|
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379 ((limb) ((s32) in[3])) * ((s32) in[9]))); |
|
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380 output[13] = 2 * (((limb) ((s32) in[6])) * ((s32) in[7]) + |
|
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381 ((limb) ((s32) in[5])) * ((s32) in[8]) + |
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382 ((limb) ((s32) in[4])) * ((s32) in[9])); |
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383 output[14] = 2 * (((limb) ((s32) in[7])) * ((s32) in[7]) + |
|
|
384 ((limb) ((s32) in[6])) * ((s32) in[8]) + |
|
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385 2 * ((limb) ((s32) in[5])) * ((s32) in[9])); |
|
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386 output[15] = 2 * (((limb) ((s32) in[7])) * ((s32) in[8]) + |
|
|
387 ((limb) ((s32) in[6])) * ((s32) in[9])); |
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388 output[16] = ((limb) ((s32) in[8])) * ((s32) in[8]) + |
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|
389 4 * ((limb) ((s32) in[7])) * ((s32) in[9]); |
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|
390 output[17] = 2 * ((limb) ((s32) in[8])) * ((s32) in[9]); |
|
|
391 output[18] = 2 * ((limb) ((s32) in[9])) * ((s32) in[9]); |
|
|
392 } |
|
|
393 |
|
|
394 static void |
|
|
395 fsquare(limb *output, const limb *in) { |
|
|
396 limb t[19]; |
|
|
397 fsquare_inner(t, in); |
|
|
398 freduce_degree(t); |
|
|
399 freduce_coefficients(t); |
|
|
400 memcpy(output, t, sizeof(limb) * 10); |
|
|
401 } |
|
|
402 |
|
|
403 /* Take a little-endian, 32-byte number and expand it into polynomial form */ |
|
|
404 static void |
|
|
405 fexpand(limb *output, const u8 *input) { |
|
|
406 #define F(n,start,shift,mask) \ |
|
|
407 output[n] = ((((limb) input[start + 0]) | \ |
|
|
408 ((limb) input[start + 1]) << 8 | \ |
|
|
409 ((limb) input[start + 2]) << 16 | \ |
|
|
410 ((limb) input[start + 3]) << 24) >> shift) & mask; |
|
|
411 F(0, 0, 0, 0x3ffffff); |
|
|
412 F(1, 3, 2, 0x1ffffff); |
|
|
413 F(2, 6, 3, 0x3ffffff); |
|
|
414 F(3, 9, 5, 0x1ffffff); |
|
|
415 F(4, 12, 6, 0x3ffffff); |
|
|
416 F(5, 16, 0, 0x1ffffff); |
|
|
417 F(6, 19, 1, 0x3ffffff); |
|
|
418 F(7, 22, 3, 0x1ffffff); |
|
|
419 F(8, 25, 4, 0x3ffffff); |
|
|
420 F(9, 28, 6, 0x3ffffff); |
|
|
421 #undef F |
|
|
422 } |
|
|
423 |
|
|
424 #if (-32 >> 1) != -16 |
|
|
425 #error "This code only works when >> does sign-extension on negative numbers" |
|
|
426 #endif |
|
|
427 |
|
|
428 /* Take a fully reduced polynomial form number and contract it into a |
|
|
429 * little-endian, 32-byte array |
|
|
430 */ |
|
|
431 static void |
|
|
432 fcontract(u8 *output, limb *input) { |
|
|
433 int i; |
|
|
434 int j; |
|
|
435 |
|
|
436 for (j = 0; j < 2; ++j) { |
|
|
437 for (i = 0; i < 9; ++i) { |
|
|
438 if ((i & 1) == 1) { |
|
|
439 /* This calculation is a time-invariant way to make input[i] positive |
|
|
440 by borrowing from the next-larger limb. |
|
|
441 */ |
|
|
442 const s32 mask = (s32)(input[i]) >> 31; |
|
|
443 const s32 carry = -(((s32)(input[i]) & mask) >> 25); |
|
|
444 input[i] = (s32)(input[i]) + (carry << 25); |
|
|
445 input[i+1] = (s32)(input[i+1]) - carry; |
|
|
446 } else { |
|
|
447 const s32 mask = (s32)(input[i]) >> 31; |
|
|
448 const s32 carry = -(((s32)(input[i]) & mask) >> 26); |
|
|
449 input[i] = (s32)(input[i]) + (carry << 26); |
|
|
450 input[i+1] = (s32)(input[i+1]) - carry; |
|
|
451 } |
|
|
452 } |
|
|
453 { |
|
|
454 const s32 mask = (s32)(input[9]) >> 31; |
|
|
455 const s32 carry = -(((s32)(input[9]) & mask) >> 25); |
|
|
456 input[9] = (s32)(input[9]) + (carry << 25); |
|
|
457 input[0] = (s32)(input[0]) - (carry * 19); |
|
|
458 } |
|
|
459 } |
|
|
460 |
|
|
461 /* The first borrow-propagation pass above ended with every limb |
|
|
462 except (possibly) input[0] non-negative. |
|
|
463 |
|
|
464 Since each input limb except input[0] is decreased by at most 1 |
|
|
465 by a borrow-propagation pass, the second borrow-propagation pass |
|
|
466 could only have wrapped around to decrease input[0] again if the |
|
|
467 first pass left input[0] negative *and* input[1] through input[9] |
|
|
468 were all zero. In that case, input[1] is now 2^25 - 1, and this |
|
|
469 last borrow-propagation step will leave input[1] non-negative. |
|
|
470 */ |
|
|
471 { |
|
|
472 const s32 mask = (s32)(input[0]) >> 31; |
|
|
473 const s32 carry = -(((s32)(input[0]) & mask) >> 26); |
|
|
474 input[0] = (s32)(input[0]) + (carry << 26); |
|
|
475 input[1] = (s32)(input[1]) - carry; |
|
|
476 } |
|
|
477 |
|
|
478 /* Both passes through the above loop, plus the last 0-to-1 step, are |
|
|
479 necessary: if input[9] is -1 and input[0] through input[8] are 0, |
|
|
480 negative values will remain in the array until the end. |
|
|
481 */ |
|
|
482 |
|
|
483 input[1] <<= 2; |
|
|
484 input[2] <<= 3; |
|
|
485 input[3] <<= 5; |
|
|
486 input[4] <<= 6; |
|
|
487 input[6] <<= 1; |
|
|
488 input[7] <<= 3; |
|
|
489 input[8] <<= 4; |
|
|
490 input[9] <<= 6; |
|
|
491 #define F(i, s) \ |
|
|
492 output[s+0] |= input[i] & 0xff; \ |
|
|
493 output[s+1] = (input[i] >> 8) & 0xff; \ |
|
|
494 output[s+2] = (input[i] >> 16) & 0xff; \ |
|
|
495 output[s+3] = (input[i] >> 24) & 0xff; |
|
|
496 output[0] = 0; |
|
|
497 output[16] = 0; |
|
|
498 F(0,0); |
|
|
499 F(1,3); |
|
|
500 F(2,6); |
|
|
501 F(3,9); |
|
|
502 F(4,12); |
|
|
503 F(5,16); |
|
|
504 F(6,19); |
|
|
505 F(7,22); |
|
|
506 F(8,25); |
|
|
507 F(9,28); |
|
|
508 #undef F |
|
|
509 } |
|
|
510 |
|
|
511 /* Input: Q, Q', Q-Q' |
|
|
512 * Output: 2Q, Q+Q' |
|
|
513 * |
|
|
514 * x2 z3: long form |
|
|
515 * x3 z3: long form |
|
|
516 * x z: short form, destroyed |
|
|
517 * xprime zprime: short form, destroyed |
|
|
518 * qmqp: short form, preserved |
|
|
519 */ |
|
|
520 static void fmonty(limb *x2, limb *z2, /* output 2Q */ |
|
|
521 limb *x3, limb *z3, /* output Q + Q' */ |
|
|
522 limb *x, limb *z, /* input Q */ |
|
|
523 limb *xprime, limb *zprime, /* input Q' */ |
|
|
524 const limb *qmqp /* input Q - Q' */) { |
|
|
525 limb origx[10], origxprime[10], zzz[19], xx[19], zz[19], xxprime[19], |
|
|
526 zzprime[19], zzzprime[19], xxxprime[19]; |
|
|
527 |
|
|
528 memcpy(origx, x, 10 * sizeof(limb)); |
|
|
529 fsum(x, z); |
|
|
530 fdifference(z, origx); // does x - z |
|
|
531 |
|
|
532 memcpy(origxprime, xprime, sizeof(limb) * 10); |
|
|
533 fsum(xprime, zprime); |
|
|
534 fdifference(zprime, origxprime); |
|
|
535 fproduct(xxprime, xprime, z); |
|
|
536 fproduct(zzprime, x, zprime); |
|
|
537 freduce_degree(xxprime); |
|
|
538 freduce_coefficients(xxprime); |
|
|
539 freduce_degree(zzprime); |
|
|
540 freduce_coefficients(zzprime); |
|
|
541 memcpy(origxprime, xxprime, sizeof(limb) * 10); |
|
|
542 fsum(xxprime, zzprime); |
|
|
543 fdifference(zzprime, origxprime); |
|
|
544 fsquare(xxxprime, xxprime); |
|
|
545 fsquare(zzzprime, zzprime); |
|
|
546 fproduct(zzprime, zzzprime, qmqp); |
|
|
547 freduce_degree(zzprime); |
|
|
548 freduce_coefficients(zzprime); |
|
|
549 memcpy(x3, xxxprime, sizeof(limb) * 10); |
|
|
550 memcpy(z3, zzprime, sizeof(limb) * 10); |
|
|
551 |
|
|
552 fsquare(xx, x); |
|
|
553 fsquare(zz, z); |
|
|
554 fproduct(x2, xx, zz); |
|
|
555 freduce_degree(x2); |
|
|
556 freduce_coefficients(x2); |
|
|
557 fdifference(zz, xx); // does zz = xx - zz |
|
|
558 memset(zzz + 10, 0, sizeof(limb) * 9); |
|
|
559 fscalar_product(zzz, zz, 121665); |
|
|
560 /* No need to call freduce_degree here: |
|
|
561 fscalar_product doesn't increase the degree of its input. */ |
|
|
562 freduce_coefficients(zzz); |
|
|
563 fsum(zzz, xx); |
|
|
564 fproduct(z2, zz, zzz); |
|
|
565 freduce_degree(z2); |
|
|
566 freduce_coefficients(z2); |
|
|
567 } |
|
|
568 |
|
|
569 /* Conditionally swap two reduced-form limb arrays if 'iswap' is 1, but leave |
|
|
570 * them unchanged if 'iswap' is 0. Runs in data-invariant time to avoid |
|
|
571 * side-channel attacks. |
|
|
572 * |
|
|
573 * NOTE that this function requires that 'iswap' be 1 or 0; other values give |
|
|
574 * wrong results. Also, the two limb arrays must be in reduced-coefficient, |
|
|
575 * reduced-degree form: the values in a[10..19] or b[10..19] aren't swapped, |
|
|
576 * and all all values in a[0..9],b[0..9] must have magnitude less than |
|
|
577 * INT32_MAX. |
|
|
578 */ |
|
|
579 static void |
|
|
580 swap_conditional(limb a[19], limb b[19], limb iswap) { |
|
|
581 unsigned i; |
|
|
582 const s32 swap = (s32) -iswap; |
|
|
583 |
|
|
584 for (i = 0; i < 10; ++i) { |
|
|
585 const s32 x = swap & ( ((s32)a[i]) ^ ((s32)b[i]) ); |
|
|
586 a[i] = ((s32)a[i]) ^ x; |
|
|
587 b[i] = ((s32)b[i]) ^ x; |
|
|
588 } |
|
|
589 } |
|
|
590 |
|
|
591 /* Calculates nQ where Q is the x-coordinate of a point on the curve |
|
|
592 * |
|
|
593 * resultx/resultz: the x coordinate of the resulting curve point (short form) |
|
|
594 * n: a little endian, 32-byte number |
|
|
595 * q: a point of the curve (short form) |
|
|
596 */ |
|
|
597 static void |
|
|
598 cmult(limb *resultx, limb *resultz, const u8 *n, const limb *q) { |
|
|
599 limb a[19] = {0}, b[19] = {1}, c[19] = {1}, d[19] = {0}; |
|
|
600 limb *nqpqx = a, *nqpqz = b, *nqx = c, *nqz = d, *t; |
|
|
601 limb e[19] = {0}, f[19] = {1}, g[19] = {0}, h[19] = {1}; |
|
|
602 limb *nqpqx2 = e, *nqpqz2 = f, *nqx2 = g, *nqz2 = h; |
|
|
603 |
|
|
604 unsigned i, j; |
|
|
605 |
|
|
606 memcpy(nqpqx, q, sizeof(limb) * 10); |
|
|
607 |
|
|
608 for (i = 0; i < 32; ++i) { |
|
|
609 u8 byte = n[31 - i]; |
|
|
610 for (j = 0; j < 8; ++j) { |
|
|
611 const limb bit = byte >> 7; |
|
|
612 |
|
|
613 swap_conditional(nqx, nqpqx, bit); |
|
|
614 swap_conditional(nqz, nqpqz, bit); |
|
|
615 fmonty(nqx2, nqz2, |
|
|
616 nqpqx2, nqpqz2, |
|
|
617 nqx, nqz, |
|
|
618 nqpqx, nqpqz, |
|
|
619 q); |
|
|
620 swap_conditional(nqx2, nqpqx2, bit); |
|
|
621 swap_conditional(nqz2, nqpqz2, bit); |
|
|
622 |
|
|
623 t = nqx; |
|
|
624 nqx = nqx2; |
|
|
625 nqx2 = t; |
|
|
626 t = nqz; |
|
|
627 nqz = nqz2; |
|
|
628 nqz2 = t; |
|
|
629 t = nqpqx; |
|
|
630 nqpqx = nqpqx2; |
|
|
631 nqpqx2 = t; |
|
|
632 t = nqpqz; |
|
|
633 nqpqz = nqpqz2; |
|
|
634 nqpqz2 = t; |
|
|
635 |
|
|
636 byte <<= 1; |
|
|
637 } |
|
|
638 } |
|
|
639 |
|
|
640 memcpy(resultx, nqx, sizeof(limb) * 10); |
|
|
641 memcpy(resultz, nqz, sizeof(limb) * 10); |
|
|
642 } |
|
|
643 |
|
|
644 // ----------------------------------------------------------------------------- |
|
|
645 // Shamelessly copied from djb's code |
|
|
646 // ----------------------------------------------------------------------------- |
|
|
647 static void |
|
|
648 crecip(limb *out, const limb *z) { |
|
|
649 limb z2[10]; |
|
|
650 limb z9[10]; |
|
|
651 limb z11[10]; |
|
|
652 limb z2_5_0[10]; |
|
|
653 limb z2_10_0[10]; |
|
|
654 limb z2_20_0[10]; |
|
|
655 limb z2_50_0[10]; |
|
|
656 limb z2_100_0[10]; |
|
|
657 limb t0[10]; |
|
|
658 limb t1[10]; |
|
|
659 int i; |
|
|
660 |
|
|
661 /* 2 */ fsquare(z2,z); |
|
|
662 /* 4 */ fsquare(t1,z2); |
|
|
663 /* 8 */ fsquare(t0,t1); |
|
|
664 /* 9 */ fmul(z9,t0,z); |
|
|
665 /* 11 */ fmul(z11,z9,z2); |
|
|
666 /* 22 */ fsquare(t0,z11); |
|
|
667 /* 2^5 - 2^0 = 31 */ fmul(z2_5_0,t0,z9); |
|
|
668 |
|
|
669 /* 2^6 - 2^1 */ fsquare(t0,z2_5_0); |
|
|
670 /* 2^7 - 2^2 */ fsquare(t1,t0); |
|
|
671 /* 2^8 - 2^3 */ fsquare(t0,t1); |
|
|
672 /* 2^9 - 2^4 */ fsquare(t1,t0); |
|
|
673 /* 2^10 - 2^5 */ fsquare(t0,t1); |
|
|
674 /* 2^10 - 2^0 */ fmul(z2_10_0,t0,z2_5_0); |
|
|
675 |
|
|
676 /* 2^11 - 2^1 */ fsquare(t0,z2_10_0); |
|
|
677 /* 2^12 - 2^2 */ fsquare(t1,t0); |
|
|
678 /* 2^20 - 2^10 */ for (i = 2;i < 10;i += 2) { fsquare(t0,t1); fsquare(t1,t0); } |
|
|
679 /* 2^20 - 2^0 */ fmul(z2_20_0,t1,z2_10_0); |
|
|
680 |
|
|
681 /* 2^21 - 2^1 */ fsquare(t0,z2_20_0); |
|
|
682 /* 2^22 - 2^2 */ fsquare(t1,t0); |
|
|
683 /* 2^40 - 2^20 */ for (i = 2;i < 20;i += 2) { fsquare(t0,t1); fsquare(t1,t0); } |
|
|
684 /* 2^40 - 2^0 */ fmul(t0,t1,z2_20_0); |
|
|
685 |
|
|
686 /* 2^41 - 2^1 */ fsquare(t1,t0); |
|
|
687 /* 2^42 - 2^2 */ fsquare(t0,t1); |
|
|
688 /* 2^50 - 2^10 */ for (i = 2;i < 10;i += 2) { fsquare(t1,t0); fsquare(t0,t1); } |
|
|
689 /* 2^50 - 2^0 */ fmul(z2_50_0,t0,z2_10_0); |
|
|
690 |
|
|
691 /* 2^51 - 2^1 */ fsquare(t0,z2_50_0); |
|
|
692 /* 2^52 - 2^2 */ fsquare(t1,t0); |
|
|
693 /* 2^100 - 2^50 */ for (i = 2;i < 50;i += 2) { fsquare(t0,t1); fsquare(t1,t0); } |
|
|
694 /* 2^100 - 2^0 */ fmul(z2_100_0,t1,z2_50_0); |
|
|
695 |
|
|
696 /* 2^101 - 2^1 */ fsquare(t1,z2_100_0); |
|
|
697 /* 2^102 - 2^2 */ fsquare(t0,t1); |
|
|
698 /* 2^200 - 2^100 */ for (i = 2;i < 100;i += 2) { fsquare(t1,t0); fsquare(t0,t1); } |
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699 /* 2^200 - 2^0 */ fmul(t1,t0,z2_100_0); |
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700 |
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701 /* 2^201 - 2^1 */ fsquare(t0,t1); |
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702 /* 2^202 - 2^2 */ fsquare(t1,t0); |
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703 /* 2^250 - 2^50 */ for (i = 2;i < 50;i += 2) { fsquare(t0,t1); fsquare(t1,t0); } |
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704 /* 2^250 - 2^0 */ fmul(t0,t1,z2_50_0); |
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705 |
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706 /* 2^251 - 2^1 */ fsquare(t1,t0); |
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707 /* 2^252 - 2^2 */ fsquare(t0,t1); |
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708 /* 2^253 - 2^3 */ fsquare(t1,t0); |
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709 /* 2^254 - 2^4 */ fsquare(t0,t1); |
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710 /* 2^255 - 2^5 */ fsquare(t1,t0); |
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711 /* 2^255 - 21 */ fmul(out,t1,z11); |
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712 } |
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713 |
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714 int curve25519_donna(u8 *, const u8 *, const u8 *); |
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715 |
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716 int |
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717 curve25519_donna(u8 *mypublic, const u8 *secret, const u8 *basepoint) { |
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718 limb bp[10], x[10], z[11], zmone[10]; |
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719 uint8_t e[32]; |
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720 int i; |
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721 |
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722 for (i = 0; i < 32; ++i) e[i] = secret[i]; |
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723 e[0] &= 248; |
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724 e[31] &= 127; |
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725 e[31] |= 64; |
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726 |
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727 fexpand(bp, basepoint); |
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728 cmult(x, z, e, bp); |
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729 crecip(zmone, z); |
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730 fmul(z, x, zmone); |
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731 freduce_coefficients(z); |
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732 fcontract(mypublic, z); |
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733 return 0; |
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734 } |
